Apex Ajax ...

To invoke Apex  you can add the following lines in your AJAX code:

<script src="/soap/ajax/15.0/connection.js" type="text/javascript"></script>
<script src="/soap/ajax/15.0/apex.js" type="text/javascript"></script>

and you can use one of the following two methods:

1- Execute anonymously via sforce.
2- Use a class WSDL.you can call the following Apex class:

global class myClass {
  webService static Id makeContact(String lastName, Account a) {
        Contact c = new Contact(LastName = lastName, AccountId = a.Id);
        return c.id;
    }
}

Here is the JavaScript code:

var account = sforce.sObject("Account");
var id = sforce.apex.execute("myClass","makeContact",
                             {lastName:"Smith",
                              a:account});
To call a webService method with no parameters, use {} as the third parameter for sforce.apex.execute.like this example:

global class myClass{
   webService static String getContextUserName() {
        return UserInfo.getFirstName();
   }
}

Use the following JavaScript code:

var contextUser = sforce.apex.execute("myClass", "getContextUserName", {});

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